Existence and Characterization of Strange Nonchaotic Attractors
in Nonlinear Systems
J. BRINDLEY
Department
of Applied
Mathematical
Studies
and T. KAPITANIAK*
and Centre for Nonlinear
YJT, U.K.
Studies.
University
of Leeds.
Leeds LS?
Abstract-Evidence
has accumulated in recent years of the occurrence, in certain nonlinear systems.
of strange nonchaotic attractors. that is attractors whose geometrical character is not simple, but on
and near to which the exponential divergence of trajectories. characteristic of chaotic behaviour. does
not occur. This behaviour has implications for predictability;
small errors in initial conditions grow
much more slowly than in a chaotic system.
Such attractors occur commonly in quasiperiodically
forced nonlinear oscillators. where their range
of existence in parameter space is substantial;
we describe two particular cases, one restricted to
mechanics. the other to chemistry. Long nonchaotic transients occur in other system.
Xloht cvidcnce for strange nonchaotic attractors arises from numerical expcrimcnts.
and certain
spectral features have been proposed [F. Romciras and E. Ott, Plrys.
Rev.
h3S. 4414 (1YS7)] as
distinguishing chsractcristics.
Some analytical methods are also indicated which give plausible and, as
compnrcd with numerical results. quite accurate bounds in parameter space for their existence.
1. INTRODUCTION
AND BACKGROUND
The existence of strange attractors
in dynamical
systems has been recognised
at lcast since
the cclcbratcd
paper of Lorcnz [l], and was suspected
well before that, though the term
itself
appears to have been used first only in 1971 by Ruelle and Takens
as a broad
description
[2]. The essential
characteristic
of a strange
attractor
is best defined
in
negatives, as for example in Romeiras and Ott [3]. Thus a strange attractor is one which is
not: a finite set of points; a limit cycle (closed curve);
a smooth (piece wise smooth)
surface; and bounded by a piece wise smooth closed surface [3-61.
This description
relates to the geometrical
character
of the attractor,
an object in phase
space towards which trajectories
are drawn as time approaches
infinity.
It says nothing
more about the trajectories
themselves,
either singularly or as a class. In particular,
it says
nothing about the rate of divergence
of two trajectories
‘initially close together’. Nevertheless, for some time, the concepts of strangeness
in geometrical
character and of exponential
divergence
of trajectories,
implying
great sensitivity
to initial conditions
or chaos, and
hence poor predictability,
were widely regarded as almost synonymous.
Recently several instances
have been described
of nonchaotic
strange attractors
[3-lo],
and, in some classes of forced nonlinear
systems, they appear to be of normal occurrence,
that is, they persist over substantial
ranges of parameter
values.
Such attractors
are
undoubtedly
strange, but the behaviour
of neighbouring
trajectories
is no longer exponentially divergent for large time even though each trajectory
may have an arbitrarily
complex
form. Thus the outlook for predicrabifity of the state evolving from a given initial condition
‘Permanent
Poland.
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Division
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323
of
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l/15.
Y%YzJ
Lodz,
J. BRINDLEY and T. KAPI~ANMK
321
is much improved;
small errors in describing
the initial stage arising, for example,
from
inaccurate
observations.
or from over-crude
approximation,
need not lead to dramaticall!,
false results in a finite time.
This difference
in qualitative
behaviour
motivates the identification
of nonchaotic
strange
attractors
(NSAs), and it is convenient
first of all to rehearse briefly the methods by which
the existence
of strange attractors,
chaotic or nonchaotic,
may be deduced.
In practice.
basic information
usually comes in the form of time series representing
the evolution
of
some set of observables.
11. as functions
of time. Conceptually
we may think of a phase
flOH
'1,(X,,)=
constituting
differential
the solution
equation
of a continuous
k = f(s).
or alternatively
map
a set of iterates,
If(f.Xl,)
system,
where
I,, of a discrete
for esample
the
autonomous
ordinary
s E W”
system.
for example
a one-dimensional
T: .r -+ f(x).
Real data, relating
values of a smooth
T + 3r..
. . T f tit,
even
even to a continuous
function,
II. obtained
.
system will of course often
at a sequence
of specific
u,,(T) --+ H,(T + r) +
--+ ll,,(T + UT) -+ .
consist
times.
of a set of
T. T + r.
.;
if this is not the case it may prove useful to generate
discrete data from continuous
tlows by suitable Poincnrc’ sections.
In casts ivhcrc the information
consists of a single time scrics, a basic step is to calculate
II frcqucncy
spectrum;
in practice this is nccomplishcd
by use of an FFT algorithm
and
appropriate
smoothing
techniques.
Knowlcdgc
of the spectrum.
togcthcr with judicious
use of PoincarG map information,
gives much insight into the nature of an attractor.
In particular,
Romcirns and Ott [3] have
suggcstcd
that NSAs yield spectra
having a chnractcristic
signature;
if the number
of
spectral components
larger than some value, B is given by i\‘(o). then h'(tr) - CT-“. This
contrasts
with values h’(a) - log (I/n),
A'(n) - [log (l/o)]’
for two-frequency
or three-frequency quasi-periodic
attractors
respcctivcly.
Marc direct evidence
of divcrgcncc
or non-divergence
of ncighbouring
trajectories
is
obtained
from the Lyapunov
exponent.
Positive
Lyapunot,
exponents
correspond
to
divcrgcncc:
they give a measure
of the rate of growth of small perturbations
to a given
trajectory,
and therefore
some measure of predictability.
It is generally
assumed that the
existence
of one or more positive
Lyapunov
exponents
is a necessary
and sufficient
condition
for the existence
of chaos.
Negative
Lyapunov
exponents
imply decay of
perturbations
and hence, in general.
convergence
of trajectories
onto attractors
of lower
dimension
than the total phase space.
Methods
of calculating
the Lynapunov
exponents,
either
from kno\vn
differential
equations
or from experimental
data, are well described
in the literature
(see. e.g. Wolf
and straightfonvardness
mean
and Swift [ 111. Wolf er nl. [ 121). and their relative cheapness
that they have been the most popular indicators of chaos. It has recently been pointed out,
by one of us (Kapitaniak
and El Naschie
[I;].
Knpitanink
[I-!]) that the
hoivevcr,
Lyapunov
exponents
derived from time series data may give misleading
results in the cxc
of NSAs and that a more scnsitivc
approach
may bc ncccssary.
In the case studied the
authors demonstrated
that the information
dimension
tl, proi,idcd a much more sensitive
test. Here
Strange
nonchaotic
attractors
325
*V(e)
dI =I’?
1(E)/ln (l/~)
with
I(E) = - 2 p[ln(p,)
1=1
where p, is the measure of the attractor in the ith ‘cube’ of a covering of the attractor by a
Cartesian
grid of spacing E, and TV
is the total number
of cubes. Effectively
p, is
obtained by estimating
the fraction of time the trajectory spends in the ith cube.
Much evidence
has now been assembled
[3-lo]
indicating
the ‘normal’ occurrence
of
NSAs is quasi-periodically
forced systems. In Section 2 we present two examples of results,
for a quasi-periodically
forced Van der Pol equation,
and for a similarly
forced system
describing an autocatalytic
chemical reactions.
Forcing
at (at least) two irrationally
related
frequencies
is common
in engineering
systems: indeed forcing at a single frequency
is likely to be the exception
rather than the
norm.
A fortiori,
in naturally
occurring
dynamical
systems,
physical
or biological.
a
multi-peaked
spectrum
of forcing is to be expected.
We have examined
in Section 2 the
sensitivity
of a nonchaotic
strange attractor,
arising from quasi-periodic
forcing, to the
presence
of a perturbatory
third-frequency
forcing.
Its robustness
suggests
that the
qualitative
behaviour
obtained
for two frequency
forcing will hold for multi-frequency
forcing, and will therefore
be observed commonly
in real systems. In particular
we expect
to see NSAs in higher order coupled systems of nonlinear
oscillators
with irrationally
related intrinsic frequencies.
Theoretical
understanding
of the phenomenon
of NSAs is incomplete,
though Romeiras
and Ott [3] have argued plausibly that a condition
for their existence is the existence of a
globally attracting three-torus
in the essentially
four-dimensional
phase space of equation:
d’x/dt’
All trajcctorics
approach
= k’ + V[costo,t
+ vdx/dr
+ sinx
this torus,
and we may expect
-t cos +I].
a dynamics
(1.1)
on the torus described
by
dcl)/dt
= S(cf), I)
Lvhcre the t dcpendencc
of S is cluasipcriodic.
The strange nonchaotic
NSAs reside on this torus (their strangeness
has been established
in the cast of the quasi-periodically
forced pendulum),
and chaos results from a fracturing
of the torus at an appropriate
parameter
value. A stroboscopic
section of these trajectories
on the three-torus
at times T, 2T, 3T,. . . nT, . . . yields a two-torus,
on which the
dynamics may be dcscribcd by an invertible
map
(1) = G(+,
r), t = (t + 2rrw/S2) mod(2n).
This map has been shown in Ref. [3] to give rise typically to NSAs.
A resemblance
is apparent
between this scenario and the similar occurrence
of two-tori
in the three-dimensional
phase space of equation
(1.1) for (oz = 0 (simple
forcing).
Quasi-periodic
bchaviour
on such two-tori might be expected to be broadly analogous
of
non-chaotic
strange behaviour
on the three-tori
above. This expectation
is born out by the
numerical
results of Section 2; the examples
exhibit a remarkably
close correspondence
between
the regions
of parameter
space occupied
respectively
by quasi-periodic
and
non-chaotic
strange behaviour
in the two cases.
2. QUASI-PERIODICALLY
FORCED
SYSTEhlS
We have alluded
in Section
1 to the ubiquity
of NSAs in quasi-periodically
forced
systems. In this section we describe two specific examples, a nonlinear
oscillator of Van der
33
J. BHI~DLEY
Pol type. and a simple autocatalytic
the first system have been described
we know, been investigated.
Qrltrsi-prriorlicol!
2.1.
We consider
forced
and T.
chemical
elsewhere
KAPITA\IA~
reaction
[8. lo]:
model [IS]. Some earlier results on
the second system has not, as far as
Van der Pol oscillutor
the equation
= (F/2)[cos(cr~ - Q)t + COS(~O+ R)r].
(2.1)
This
equation.
dcscribinp
/j > 0. implying
in innumerable
Forcing,
when
real
systems
an oscillator
the esistence
models
included.
endure
cxnmplc of this.
Detailed
numerical
with
of robust
nonlinear
finite
amplitude
damping.
has a stable
free oscillations.
of NSAs
forcing
at two
evidence
indicated;
cycle
for
used
of mechanical.
electrical
has usually been assumed
and chemical
and biological
systems.
to be at a single frequency.
Incvitnbly.
and equation
(2. I) is a simple
frequencies,
or more
of the existence
of an NSA
in this system
elsewhcrc
[S. 101. and its range of esistcnce in (F. 52) space for given values
is sholvn in Fig. I. The corresponding
largest Lynapunov
csponcnt
is shown
regions
limit
It has been
in these
regions
RIO positive
cxponcnt
esists,
I 33 x 106 ,
1400
1600
f?Qi:
has been
given
of J.. /;. ~rj,,, (I)
in Fig. 2. \vith
but the behnviour
Strange
nonchaotic attractors
is not quasi-periodic.
Not only is this deducible
analytically
[9, lo] that no response of the form
x = Acos(wt
from
327
the numerics:
we have
established
+ E~)cos(~-~~+ E:)
(24
is possible.
Note that the region of existence of NSA is extensive in the parameter
space (Fig. 3). so
that we should expect to observe such behaviour
in a real system modelled
by equation
(2.1).
As a test of the robustness
of the behaviour
to further complexities
in the forcing, we
have tested the response of the system when a third irrationally
related frequency
is added
to the forcing function. The results are shown in Fig. 4, and relate to the equations
.i: - 2;1(1 - .+
+ x = acosntcostur
+ bcosrnr
(2.3)
and
.i: - 2L(1 - x?).t
+ x = acoscu,tcosRtcosor
(2.4)
respectively.
In each case regions of existence of a nonchaotic
strange attractor
survive, and indeed,
with three frequencies,
even when the third frequency
arises at the level of the small
perturbation.
we find that it is impossible
to have anything
other than chaos or NSA. In
the case where b < N the power spectrum is typically like Fig. 5, suggesting a ‘noisy’ torus,
on which neighbouring
trajectories
do not diverge exponentially.
WC conjecture
that such a
noisy quasi-periodic
bchaviour
will be the typical response
of a real system to quasiperiodic forcing.
EXISTENCE OF TWO FRCO,,ENCy
~ASIPERIOOIC
SOLUTION
jjzjggj!
$1
:I
2.9
Fig.
3.
Domains
of strange
Fig. 4. The Inrgcs: nonzcw
two frcqucncy forcing:
chaotic
and
strange
CHAOS
I
I
3.0
nonchaotic
32
34
attractors
Lyapunov
cxpcrimcnt
A,.,, for equations
(7~ = 0. h = 0: (b) equation
(2.3):
6 = d/3/10.
h
x 103
of equation
(1.1):
F = 1.1 X IO”.
R =
15(H).
(2.3) and (2.4): A= 2.5. 0 = 4. 52= 2.64: (a)
b = 0.1; (c) equation
(2.4) 0 = ti3jlO.
J. BRINDLEY and T. KAPITANIAK
0.01 0
1
3
2
FREQUENCY
Fig. 5. Power
2.2.
Our
specrrum
of
the
response
of equation
(2.2):
h = I). I.
A = 2.5.
a = 5.
R = 2.6-I.
\/Z/IO.
rr/ = V?
IO.
~lrnsi~~criotlictrl!\,Jorced drerrlicnl spstcrn
second
is a chcmicnl
esample.
system.
bvhich,
because
WC consider
of its relative
a model
for cubic
unfamiliarity
we describe
autocatalysis,
developed
Scott [lS], for a simple reaction
in a closed vessel which converts
product C via the intermediates
A, B. according
to the scheme
A + 2B -
313
B-C
In the
type
orders
of chemical
of
system
magnitude
the
more
fully.
by Gray
reactant
and
P to the
rate = k,,p
P-A
many
w-
envisngcd
grater
than
rate
= k,clh’
rate
= k7/).
(2.5)
the concentration
the
masimurn
uf the
concentrations
initial
reactant
attain4
by
1’ i5
the
intermccliatcs
A. B. with (to bc consistent)
the rate of convt‘rsiun
from P to A being
wc can. to
rclativcly
slwv in comparison
with the other reaction rata
k,. X-,. Conscqucntly
regard the concentration
P as being constant and equ;ll to its initial
;I good approximation,
value
/I,,.
i.e. LVL‘art‘ making
The diffcrcntial
equations
the ‘poolctl
governing
chemical’
approximation.
the reaction
scheme
drr/dr’
= k,,/‘,, -
dh/d/’
= ,&r/1)
then become
k,ntG
(2.6)
where
made
c~h are
the concentrations
non-dimensional
kzh
respectively
and
I ’ is time.
Equations
(2.S)
are
by Lvriting
0 = .r(k,/k,)“.
so that equations
of AB
-
(2.6)
h = y(k,jk,)”
and
f’ = kzf
become
(2.7)
whcrc
!’ = (k,,p,,/k,)(
k ,/k,)’
z
is n constant of order unity.
Equations
(2.6)
\vcrt’
originally
proposed
b>, Sel’kov
[I61 as a simple
model
for
oscillations
in glycolysis.
and their solution
has been consideral
for all /l > 0 in some detail
in Ref. (IS]. Strictly,
equation
(2.5) implies
that p decays very slwvly
requiring
!l to be
Strange nonchaotic
attractors
329
proportional
to exp (-at)
(CY> 1). This scheme has been studied by Merkin et al. [IS]
where a full discussion of the application
of such a scheme to chemical reactions is given.
Our basic assumption
is to set cr to be zero.
If we assume ,D to be oscillating quasi-periodically
about ,LJ,)with frequencies
W. R, in an
attempt
to model the effects on the reaction
due to fluctuating
external
conditions.
we
consider the system
d.r/d t = &j(l
dy/dt
+ Ecosnt
cos or) -
_r_$
(2.8)
= xy’ - y.
A typical set of results is shown in Fig. 6, which should be compared
with Fig. 7
obtained
by Merkin et al. [19] in a study of the simple sinusoidally
forced reaction (i.e.
forcing function proportional
to cos~t).
Again note the close, but not exact, correspondence between
regions of quasi-periodicity
in the simply forced oscillator
and regions of
existence of an NSA in the quasi-periodically
forced case.
Cl2
ESCPPE
STRANGE
TO ININITY
NONCHAOTIC
OUASIPERIODIC
012
ESCAPE
TO INFINITY
ouw_slPmloolc
II
Fig.
7. Domains
of chaotic (cross-hatched)
and qunsipcriodic
behaviour
of equation
(2.8):
14, = 0.95,
w = 0.
330
J.
3. TRANSIENT
STRANGE
BRINDLEY
and T.
KAPITANIAK
NON-CHAOTIC BEHA\‘IOUR
SYSTEMS
IN PERIODICALLY
FORCED
We have seen that NSAs are common in quasi-periodically
forced systems. In this section
we show that transient strange nonchaotic
behaviour
can occur also in periodically
forced
systems.
Consider the parametrically
excited Duffing equation
.i- + a.? - (1 + b cos Rr)s
+ cx3 = 0
(3.1)
where n, 6. c and S2 are constant
(Ariaratnan
et al. [19]). Examples of this equation
are
found in many applications
of mechanics,
particularly
in problems
of dynamic stability of
elastic systems.
Equation
(3.1) has three Lyapunov
exponents:
one of them is always 0, one is aluays
negative, and the third can change sign with the change of system parameters.
This one can
be called the largest non-zero
Lyapunov
exponent,
i.. It is plotted in Fig. 3.1 as 6 changes
from 0 to 0.5. When h is small. /! is negative and the system (3.1) does not show sensitive
When b is about 0.34.
2 changes suddenly
from
dependence
on the initial conditions.
negative to positive values and the behaviour
of the system becomes chaotic.
The winding number fulfils the relation
)t’ = lim [a( I) ,-r
0(0)1/f
= (I/n)R
(3.7)
where:
(_v, _c) = (rcosn.
r sin a):
I, II arc intcgcr
only, or 0 i 0.256. In the intorvnl
0.256 < h < 0.3JS we have aperiodic
motion without sensitive
dcpcndance
on the initial
conditions.
which wc call transient
strange non-chaotic
bchaviour.
Since in 3-dimensional
phax
space
the
combination
of Lynapunov
esponents
(O.-.-)
guarantees
ultirnatc
approach
to a limit cycle, so the observed
bchaviour
has to bc transient.
This transient
behaviour
differs frorn chaotic transient
bchaviour
in that the nearby orbits do not divcrgc
csponcntially.
Time evolution
towards a limit cycle seems to bc following
chaotic
transient
A> 0
+
strange non-chaotic
transient
1. < 0
--L limit cycle.
Strange nonchaotic
331
attractors
The Poincare
maps for the parameter
value close to the boundary
between
transient
strange nonchaotic
behaviour
and chaotic attractors are shown in Fig. 9(a,b).
The system (3.1) has three equilibrium
positions,
at x = kl, 0 and i = 0. Depending
on
the initial conditions
it can exhibit oscillations
around one of the two stable equilibria
x = 21. i = 0, a small ‘orbit’. or around all three equilibria,
a large ‘orbit’ (see Fig. 10).
In Fig. 11 we show the plot of maximum deflection
X from equilibria
x = _Cl, i = 0 in
the case of motion on the small orbit. and from equilibrium
x = 0, f = 0 in the case of
motion on the large orbit.
For the initial conditions
leading to the oscillations
on the small orbit it is found that this
type of oscillation
exists up to 6 = 0.308, when we have a sudden transition
to the motion
(b)
Fig. 9. PoincarC maps: (a) transient.
X
I = lO-‘T, h =
0.34; (b) 6 = 0.35.
33:
J. BKINDLEY and T. KAPITA.SIA~
(b)
(a1
x
Collusion typeII
08
07
t
06
Transllmn IO
chaos
05
04
03
02
arouncl
three
with
the
this
cvcnt
equilibria
collision
;t
of
type
with
the
a long
small
apcrioclic
orbit
I collision.
with
at
Nest.
transient.
;1 non-stable
h = 0.33X,
This
~uclclcn
orbit
around
transition
is conncctccl
.v = 0. .\‘- = 0.
LVC obs2rvc
;I
transition
Ii’C
to
cdl1
chaotic
bchaviour.
For
the
stable
only
for 0.2-G
> (7 > 0.256.
st:lhlc.
At
11 = 0.256
WC observe
around
.Y = 0.
non-chaotic
initial
conditions
.i = 0.
for
;I type
bchaviour.
As
which
For
the
large
orbit
h > 0.2-H.
the
only
collision
and
collision.
II
in the
first
cabc.
for
of
for
wc’ find
is possible.
the
the
oscillations
large
larger
orbit
17 ac
!? S 0.3JS.
that
on
the
Lvith
the
ohscrxc
mc uhscrvc
this
orl,it
orbit
small
unstahlt‘
lransicnt
transition
ih
arc
orhit
5tr;ingc
to chaotic
bchaviuur.
b’c
can
12.
As
;iIso
fis
the
and change
attractor.
(1 incrcascs
C(= 0. LSL
;I collision
the
value
of
vnlucs
from
0.1
of
type
h
to
corrcsponcl
of a. The
plot
to O.lSl
Lvc ohscrve
I occurs
and
to
motion
on
of dutlcction
,Y
pcridic
rnotiun
tramient
strange
;I strange
versus
un
norlchaotic
non-chaotic
f/ i5 sho\vn
the
small
txha\
iour
in Fig.
or-hit.
At
t’n\ul’.
Strange
nonchaotic
attractors
333
0.6
Fig. 12. Maximum
this
esisting
return
4.
deflection
up to n = 0.212,
when another
to periodic oscillations
II)ENTIFIC/\TION
X vs damping
coefficient
u for equation
type I collision
(3. I).
takes place, and we have a
on the small orbit.
AND
I’KEI)ICTION
OF NONCHAOTIC
STRANGE
ATTHACTOHS
The results WC have presented in earlier sections have all been obtained, as have those of
other workers, by direct numerical computation of indicators. Markedly different spectral
characteristics scparatc NSA behaviour from quasi-periodic bchaviour, and calculation of
Lyapunov cxponcnts provides further evidence, reliable when the basis of calculation is a
known mathematical system, but to bc used with care when the only source of information
is actual data. In this latter case, which is of course likely
to be the usual one in situations
of real physical concern, secure evidence is obtainable only by use of much more
computationally demanding dimension calculations.
The practical indications for predictability associated with the existence of NSAs makes
some form of analytical prediction highly desirable; we have found that two approaches
yield promising results (8, lo].
Firstly we can find limits on parameter values for the existence of a purely quasi-periodic
response to quasi-periodic forcing. The method uses a theorem first proved by Urabe [20],
and an example of the resulting predicted boundary in parameter space is given in Fig. 13.
Briefly
the theorem quarantees the existence of a quasi-periodic solution (2.2) of the
equation (2.1) if the inequality
C s
is satisfied,
where the constant,
(1/52;)@){(0$
-
A?)/(2 + 2A)}‘+
(4.1)
C is given by
C = max{(F/2)/1(0;
(F/2)(R
-
-
(Q -
o)?/ + (F/2)/jW?,
W)/J(c)(?l
- (Q -
+
+ (F/2)(Q
-
(a + W)zj,
+ cu)/\W; -
(Q + tu)‘l.
This result may bc extended to establish conditions for the existence of higher harmonic
quasi-periodic responses as follows. If the condition (4.1) does not hold, i.e. if there is no
simple quasi-periodic
solution
of the form (2.2),
we can expect more complicated
J. BRI~DLEY and T. KAPITASIAL
33-J
Fig, 13. Boundnry
of existence
quasi-periodic
solutions
.~(f)
of the solution (2.2) (broken line) and houdar~~~
of Hopf
for equation (2.1). (system parameter\
;I+ in Fig. 1).
([I,
icients
1,) =
L?(C), (1).
I’,“,
i3(0. 0)
=
C C {C,,cos(p,
+
11’1
= ll’ll + lP2I.“I
/‘zl’:,
+
C,,[l,,
can
be
_v( [) =
U(0.
0) +
the residual
R(f)
Expanding
R(f)
Jj,sin(p. 19f}
v)f +
=
?
-
by
c
c
r=, I,“=’
{C,, cos(/~.
26(1
-
/3.C’).?
in finite
cloublc
Fourier
R(f)
b(0,
+
=
0)
1
,?,
any
approsimatc
v)t
+
fi,,sin([j,
+
scrics
x
F(cosv,f
to;bT -
+
method
introduce
and
then
WC hnvc
v)f}
{c,,cm((p,
v)f
+
rl,,sin(p.
I,‘!”
the incqualitics
O)l +
cos\~lf).
one obtains
Jr,,
lh(O,
(4.2)
= !2 - to, vz = S2 + (0, the unknown
dctermincd
function
r =
If we now
line)
Ip,=r
Galcrkin).
WC have
(wlid
of the form
r-l
where
hifurcatwn
1
1
,=* I,‘!”
{lc,l
+
!ti,,/)
v)L}
(for
coeffeXampie
Strange
T
nonchaotic
2
attractors
335
sup Ii(t)1
T’ a sup Ix’(r)\.
If the solution
z lies in a &neighbourhood
Il’P(:. A) -
Taking
number
A@)[\ s 2A{7-(2T’
into account (4.3) we have
K< 1, and a positive number
2A{T(27-’
of i(t)
B(t)]’
we have
+ 7’) + 2(r’ + 2T)6
the following
results:
if there
6 satisfying both inequalities
+ T) + 2(7-’ + 2776
2r(2
= [i(t),
+ 3S9)
+ 2@‘,‘/A(l
< (/I/2)((1
- K)(l - P)‘R
+ 36’)
exists
- A9’%/2(2
(4.3)
a non-negative
+ 2A)‘r-
< d
then by the theorem of Urabe, the quasi-periodic
solution (4.2) exists.
A second analytical condition,
which apparently
gives an approximation
to the boundary
between
nonchaotic
and chaotic behaviour,
is obtained
by seeking Hopf bifurcations
in a
simpler
system
of ordinary
differential
equations
obtained
from (2.1) by a suitable
averaging
procedure.
The details have been fully described
elsewhere,
and typical results
are included in Figs 3 and 13.
Aside from these two partly analytical
results, detection
of NSAs has been achieved
through direct numerical
analysis of data: calculation
of spectra or calculation
of some
derived quantity,
for example
either Lyapunov
exponent
or some form of information
dimension
is required.
Romeiras
and Ott [3] have proposed
a method
based on direct analysis of spectra.
Introducing
N(a), defined as the number of spectral components
larger than some value u,
they conjecture
that N(a) - o-”
for NSAs in contrast
to N(a) - In (l/b)
for two
frcqucncics
quasi-periodic
attractors
and N(a) - In’( l/a)
for three frcqucncics
quasiperiodic attractors.
Evidcncc for such distinctive
spectral behnviour
in the forced damped
pendulum
equation was adduced by them.
An altcrnativc
approach,
based on Lyapunov
exponents.
has been proposed by Ditto et
crl. [21] and Kapitaniak
[l-t]. Estimation
of Lyapunov
exponents
is rcliablc
when the
equations
driving the system arc known but is not reliable when used on data obtained
from a long time scrics of observations
in a case where the eqlrnfiorrs arc unknown.
In this
second
case, in order to distinguish
between
chaotic
and non-chaotic
attractors,
the
properties
of information
dimension
d, are used. The presence
of two-frequency
quasiperiodic forcing guarantees
that every attractor will be at least two-dimensional.
According
to Kaplan-Yorke
conjecture
[22] the information
dimension
of strange nonchaotic
attractors is d, = 2. In practice the dimension
estimation
is performed
on the surface of cross
section of the attractor,
reducing all dimensions
by one, i.e. a strange nonchaotic
attractor
occurs if the information
dimension
of an attractor
on the cross section is tl, = 1 and a
chaotic attractor occurs if d , 2 2. More details of this method are given in Ref. [14].
5. DISCUSSION
AND
SIJM~IAKY
Our objectives in this paper have been threefold.
Firstly to draw further attention
to the
occurrence
of a type of behaviour
in nonlinear
dynamical
systems which is complex, but in
which neighbouring
trajectories
do not diverge
exponentially,
as they do when the
behaviour
is chaotic. We have attributed
this behaviour
to the existence
to a nonchaotic
strange attractor
(NSA). Secondly we have stressed the robustness
of such a behaviour,
particularly
in systems which arc quasi-periodically
forced, and thirdly we have described
336
J. BRI~DLEY
and T. KAPITwI-\h
methods for the detection
of NSAs and for the establishment
of bounds in parameter
space
for their existence.
The distinction
between
an NSA and a strange attractor
is likely to be important
when
detailed
calculation
of trajectories
is required.
Though
trajectories
may appear equalI)
complex in the two cases. small errors in an initial condition
will remain small for far
longer if we have an NSA. and we should expect much better predictability.
Though
transient
non-chaotic
strange behaviour
is possible in periodically
forced systems, it seems
that permanent
NSAs occur widely for systems which are forced quasi-periodically
and that
they persist
when
the forcing
has still more
independent
frequencies.
Indeed.
the
parameter
space for a system forced at multiple
frequencies
may well be divided into
regions of chaotic and nonchaotic
strange behaviour:
generation
and analysis of suitable
data will be valuable.
Finally. several avenues
of further investigations
suggest themselves.
Of much interest
will be the examination
of coupled systems of nonlinear
oscillators having different intrinsic
frequencies,
and of (formally
infinite-dimensional)
fluid systems with strong modal structure forced by boundaries.
We might expect to find NSAs at least in cases Lvhere one or
t\vo oscillators
dominate.
effectively
driving
the others
much as the forced systems
considered
here. Clearer understanding
of several of the ‘analytical’ results is also awaited,
especially of the spectral signature
of NSAs, introduced
by Romeiras
and Ott [3]. and of
the importance
of the Hopf bifurcation,
in the averaged system for the onset of chaos in
the original system. which WC have dcscribocl in Section 4. Above all, as in many fields of
suites of numerical
experiments
are vital
dynamical
systems theory, further well planned
for the establishment
of a database against which to test thcorctical
ideas.
Finally one should mention
a result which was recently
rcportcd
by Kapitaniak
[l-h].
hscd on ;I gencralizntion
of the triadic cantor set to higher dimension
El Naschic showed
that qu;G-crgodicity
is typical for four dimensional
dynamical
systems. The implications
of
this result for cltl;lsi-pcriuclic~llly
for4
oscillator-5 and strange nonchaotic
sets arc obvious.
Strange
nonchaotic
attractors
337
15. P. Gray and S. Scott. A new model for oscillatory
behaviour
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Phps. C/rem.
90. 985 (1986).
16. E. E. Sel’kov. Self-oscillations
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17. J. Merkin. D. Needham
and S. Scott, On the creation.
growth and extinction
of oscillatory
solutions for a
simple forced chemical reaction scheme. S/AM 1. Appl. Marh. 17, 1040 (1987).
1s. J. Merkin. D. Needham
and S. Scott, Oscillatory
chemical reactions
in closed vessels. Proc. R. Sot. Lo&
A406. 299 (1986).
19. S. T. Ariaratnam.
W. C. Xie and E. R. Vrscay. Chaotic motion under parametrical
excitation.
Dynom. Sfabil.
Swms
4. 11 (1989).
20. ht. Urabe. Quasiperiodic
solutions of ordinary differential
equations,
Nonlin. Vibr. Probl. 18, 85 (197-I).
21. W. L. Ditto. bl. L. Spano. H. T. Savage, S. W. Rauseo. J. Heagy and E. Ott, Experimental
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strange nonchaotic
attractors,
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Chuotic Oscilhions
in Mechnical
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